Hypothesis testing123456789101121314151617.pptx

statistics

1. Statistics

2. Hypothesis Testing Hypothesis testing is a fundamental concept in statistics used to make inferences or draw conclusions about a population based on sample data. Here’s a basic introduction to the process.

3. What is hypothesis testing..? Hypothesis testing is a method used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves comparing data against a null hypothesis to see if there is enough evidence to support an alternative hypothesis. A hypothesis test provides us with a statistical method so that we can use our sample findings to make an inference [conclusion] about a population The aim is to rule out sampling errors.

4. Hypothesis Testing Null Hypothesis (H₀) : This is a statement of no effect or no difference, and it represents the default or baseline position. It is what you assume to be true unless evidence suggests otherwise. Example: If you are testing a new drug, the null hypothesis might be that the new drug has the same effect as the current standard drug. Alternative Hypothesis (H₁ or Ha) : This statement is what you want to test for. It represents a new effect or difference that you suspect might exist. Example: The alternative hypothesis could be that the new drug is more effective than the current standard drug.

5. Hypothesis Testing Steps in the process: State hypothesis [what do we think is happening in the population?] 2. Use this hypothesis to predict the characteristics the sample should have [ e.g direction of change] including the ‘critical region’ 3. Obtain a sample from the population and calculate the sample statistic 4. Compare the sample data findings to the hypothesis to reach a conclusion

6. Hypothesis Testing Steps in the process: State hypothesis [what do we think is happening in the population?] Null and Alternative hypotheses 2. Use this hypothesis to predict the characteristics the sample should have [ e.g direction of change] including the ‘critical region’ Directional – direction of change [one-tailed test] Non-directional – think something has changed, but not sure how [two-tailed test] Alpha level [∝ ] to decide critical region [unlikely to occur if Null Hypothesis is true] 3. Obtain a sample from the population and calculate the sample statistic Collect some data! Introduction to z -statistic [one-sample z-test] 4. Compare the sample data findings to the hypothesis to reach a conclusion Is there sufficient evidence to convince us …

7. One – Tailed test A one-tailed test in hypothesis testing is used when you are interested in detecting an effect or difference in only one direction. This type of test evaluates whether a sample statistic falls into one specific tail of the probability distribution, based on the alternative hypothesis. A one-tailed test is appropriate when you have a specific direction of interest. It can test for either an increase or a decrease but not both. Left-Tailed Test : Null Hypothesis (H₀) : The parameter is greater than or equal to a certain value. For example, H₀: μ ≥ μ₀. Alternative Hypothesis (H₁ or Ha) : The parameter is less than that value. For example, H₁: μ < μ₀.

8. Hypothesis testing – uncertainty / errors Hypothesis testing is an inferential process Based on limited information to reach a conclusion That is, a sample provides limited / incomplete information about the population, and yet a hypothesis test uses a sample to draw a conclusion about the population Therefore, there is always the possibility that an incorrect conclusion will be made … Two types of errors: Type I error (False Positives) Type II error (False Negatives)

9. Type I and Type II errors Type I error: – Rejecting when it is true • suggesting a difference when there is NO difference [False Positive] Type II error: – NOT rejecting when it is false • suggesting NO difference when there IS a difference [False Negative]  

10. Hypothesis testing – Type I error – False Positives A Type I error occurs when a researcher rejects a null hypothesis that is actually true concludes that a treatment does have an effect / there is a difference / relationship when in fact it has no effect / there is no difference / relationship A Type I error occurs when a researcher unknowingly obtains an extreme, non-representative sample Our hypothesis tests are structured to minimise the risk that this type of error will occur The alpha level for a hypothesis test is the probability that the test will lead to a Type I error the alpha level is the probability of obtaining sample data in the critical region even though the null hypothesis is true. If ∝ = .05, then around 1/20 studies will show a significant effect, when it shouldn’t. As studies are assumed to require independent replication, we accept this error rate

11. Hypothesis testing – Type II error – False Negatives A Type II error occurs when a researcher fails to reject a null hypothesis that is actually false concludes that there is no effect / difference / relationship when there really is one A Type II error occurs when the sample mean is not in the critical region even though there is an effect / difference / relationship Usually occurs when the effect / difference is small The consequences of a Type II error are usually not as serious as a Type I error A Type II error suggests that the means do not show the results expected / hoped for by the researcher

12. Type I and Type II errors   Do Not Reject H Reject H H is TRUE H is FALSE Correct Pregnant Correct Not Pregnant Type I Error (False +’ve) Type II Error (False -’ve)

13. Selecting an Alpha level The main concern when selecting an alpha level is to minimi z e the risk of a Type I error alpha levels tend to be very small probability values usual convention is that the largest permissible value is ∝ = 0.05 However … as the alpha level is lowered [e.g., to 0.01 / 0.001] the hypothesis test demands more evidence from the results and the risk of making a Type II error increases

14. One – Tailed test Right-Tailed Test: Null Hypothesis (H₀): The parameter is less than or equal to a certain value. For example, H₀: μ ≤ μ₀. Alternative Hypothesis (H₁ or Ha): The parameter is greater than that value. For example, H₁: μ > μ₀.

15. One – Tailed test Example 1: Right-Tailed Test Scenario : A manufacturer claims that a new battery lasts 500 hours. You want to test if the battery lasts longer than 500 hours. Null Hypothesis (H₀) : The mean battery life is less than or equal to 500 hours. H₀: μ ≤ 500. Alternative Hypothesis (H₁) : The mean battery life is greater than 500 hours. H₁: μ > 500. You perform a test, calculate the test statistic, and compare it to the critical value for the right tail. If the test statistic falls in the right tail of the distribution and the p-value is less than α (e.g., 0.05), you reject the null hypothesis, suggesting that the battery life is significantly greater than 500 hours.

16. One – Tailed test Example 2: Left-Tailed Test Scenario : You are testing whether a new diet pill results in a reduction in average weight loss compared to a standard diet pill. The standard pill has been reported to cause a weight loss of 10 pounds. Null Hypothesis (H₀) : The mean weight loss with the new pill is greater than or equal to 10 pounds. H₀: μ ≥ 10. Alternative Hypothesis (H₁) : The mean weight loss with the new pill is less than 10 pounds. H₁: μ < 10. You perform the test, calculate the test statistic, and compare it to the critical value for the left tail. If the test statistic falls in the left tail and the p-value is less than α, you reject the null hypothesis, indicating that the new pill results in significantly less weight loss than the standard pill.

17. Two – Tailed test A two-tailed test in hypothesis testing is used when you want to determine if there is a significant difference or effect in either direction from the null hypothesis. Unlike a one-tailed test, which focuses on detecting changes in a specific direction, a two-tailed test evaluates both possibilities whether the observed effect is either significantly greater or less than the hypothesized value. A two-tailed test is appropriate when you want to test for deviations in both directions from the null hypothesis. It evaluates whether the sample data falls significantly in either tail of the distribution.

18. Two – Tailed test Null Hypothesis (H₀) : The parameter of interest is equal to a specified value. For example, H₀: μ = μ₀, where μ is the population mean and μ₀ is a hypothesized value. Alternative Hypothesis (H₁ or Ha ): The parameter of interest is not equal to the specified value. For example, H₁: μ ≠ μ₀. The significance level (α) is the probability of rejecting the null hypothesis when it is true (Type I error). In a two-tailed test, the significance level is split between the two tails of the distribution. For a significance level of 0.05, the critical region is divided into two tails, each with 0.025 (totaling 0.05).